Characterizing certain incomplete infinite-dimensional absolute retracts.

*(English)*Zbl 0629.54011Let \(\sigma =\{t\in [-1,1]^{\infty}| t_ i=0\) for all but finitely many \(i\}\) and let \(\Sigma =\{t\in Q^{\infty}| t_ i=0\) for all but finitely many \(i\}\) where \(Q=[-1,1]^{\infty}\). The second author has obtained characterization theorems for \(\sigma\) and \(\Sigma\) [Proc. Am. Math. Soc. 92, 111-118 (1984; Zbl 0577.57005)]. The authors obtain characterizations for \(\sigma\) and \(\Sigma\) as corollaries of a more general characterization theorem for certain incomplete subsets of Hilbert spaces. This more general characterization theorem also yields a characterization of \(\Sigma\times \sigma\) and gives characterizations of certain absolute Borel sets and of certain ARs. A triangulation theorem for the spaces characterized is also obtained. One version of a general characterization theorem obtained is as follows. If \({\mathcal C}\) is an additive topological class hereditary with respect to closed subsets, and if there exists a \({\mathcal C}\) absorbing set \(\Omega\) in a Hilbert space, then an AR X is homeomorphic to \(\Omega\) if and only if X is \({\mathcal C}_{\sigma}\), X is strongly \({\mathcal C}\) universal and \(X=\cup^{\infty}_{i=1}X_ i\) where each \(X_ i\) is a strong Z-set in X. X is strongly \({\mathcal C}\) universal if for each map f from C to X, where \(C\in {\mathcal C}\), for each closed subset D of C such that \(f|_ D\) is a Z-embedding, and for each open cover \({\mathcal U}\) of X, there exists a Z-embedding h from C into X such that \(h|_ D=f|_ D\) and such that f and h are \({\mathcal U}\) close.

Reviewer: D.J.Garity

##### MSC:

54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |

54F65 | Topological characterizations of particular spaces |

54C50 | Topology of special sets defined by functions |